Rationally connected varieties over finite fields

Let X be a geometrically rational (or, more generally, separably rationally connected) variety over a finite field K. We prove that if K is large enough, then X contains many rational curves defined over K. As a consequence we prove that R-equivalence is trivial on X if K is large enough. These results imply the following conjecture of J.-L. Colliot-Thélène: Let Y be a rationally connected variety over a number field F. For a prime P, let Y_P denote the corresponding variety over the local field F_P. Then, for almost all primes P, the Chow group of 0-cycles on Y_P is trivial and R-equivalence is trivial on Y_P.